The basic properties of stars we want to know are their:

  Distance
  Luminosity
  Radius
  Mass
  Surface Temperature
  Interior Temperature
  Chemical Composition
  Internal Structure
  History and Age
  Planetary companions

 Stars go through their life cycles on timescales much longer than a
 human lifetime; in order to figure out what's going on, then we look
 at lots of different stars and try to use this to determine the life
 cycles of typical stars.

The brightness B is the quantity of energy given off by an object per
unit time, per unit area of the detector we use to detect it.  Imagine
an enormous sphere centered on a distant star reaching out to us a
distance d away; the light from that star is spread out over the
surface of the entire sphere.  This gives us the relationship between
brightness and the luminosity L of the star (the 'inverse square law'):
  B = L/(4 pi d^2)

Thus with a measurement of the brightness of a star, and of the
distance via the inverse square law, we can determine the luminosity.

 We measure distance via parallax: the apparent change of direction
pointing to a star as the Earth goes from one side of its orbit around
the Sun to the other.  Half of this angular change is called parallax angle p:

 p = (1 AU)/distance

where p is measured in radians (2 pi radians = 360 degrees).
There are 200,000 arcseconds in a radian.  The nearest stars are about
4 light years (300,000 AU) away, for which the angle p is 0.75
arcseconds; that is much too small to measure without a telescope.
Compare with the Sun, which is 8 light minutes away.  Confusingly, the
so-called parallax angle in the specific situation of the change of
position of a star as the Earth moves around the Sun is defined to be
*half* the full anglular motion over 6 months 
(i.e., the angle subtended by 1 AU, not 2 AU, which is the baseline 
for 6 months of the orbit). Of course, note that the parallax concept 
is broader than just the
motion of the Earth around the Sun.

 A "parsec" is the distance at which a star has a parallax of 1
 arcsec; it turns out to be about 3.3 light years.

 To first approximation, stars radiate as blackbodies.  The hotter a
star, the shorter the wavelength at which the radiation peaks by the
Wein Law.  Cool stars (say, 3000K) appear quite red, and hot stars
(50,000K) appear blueish.  The Sun is in the middle, 5800K, and
appears white (not yellow!  So the measurement of the spectrum of a
star tells us its surface temperature.

Stars are given different types depending on their surface
temperature, and therefore their color:
O B A F G K M L (from hottest to coolest).  The Sun is a G star.
 This scale was put together by Annie Jump Cannon.

 The spectra of stars show the overall shape of a blackbody, but also
have superposed absorption lines.  Atoms in the atmospheres of the
stars selectively absorb photons of specific wavelengths, as the
electrons in them make transitions from one energy level to another.

A bunch of new concepts: the electrons in an atom can exist at only
specific energy levels.  When it makes a transition from one level to
another, it gives off energy in the form of electromagnetic
radiation.  This radiation is packaged into a *photon*, in which the
light has a particle nature.  (Remember we learned that light is a
wave?  Turns out that you can't divide up the wave into arbitrarily
small chunks of energy).  Light of a given wavelength or frequency nu
corresponds to a photon of a specific energy.  The relationship is:
 Energy = h nu

where h, Planck's Constant is roughly 6 x 10^{-34} Joule seconds.

Each type of atom absorbs its own specific wavelengths corresponding
in energy to the difference between two energy levels in that atom; by
recognizing these wavelengths, we can infer the chemical composition
of the star.  The story is complicated that certain transitions in
elements can happen only when the temperature is just right.  For
example, hydrogen atoms are primed to absorb photons in the visible
part of the spectrum only when the temperature is between roughly 8000
K and 15,000 K (cooler than this, hydrogen forms the H_2 molecule
which has no absorption features in the visible part of the spectrum,
and hotter than this, the hydrogen becomes ionized).  This means that
absorption lines of hydrogen are not present in the spectra of cool
stars, even if there is a lot of hydrogen in the atmosphere.  Cecilia
Payne-Gaposchkin was the first to sort this all out, and came to one
of the most profound conclusions in the history of astronomy: Stars
are made almost exclusively of hydrogen and helium; essentially all
stars have the same chemical composition.

Being quantitative, stars are 74% Hydrogen, 24% Helium, and 2%
everything else: the next dominant elements are Oxygen, Carbon, Neon,
Iron, Nitrogen, Silicon, and Magnesium, in that order.  These values
are counting atoms by mass.

Enjar Hertzsprung and Princeton's own Henry Norris Russell
independently made the 'Hertzsprung-Russell' (or HR) diagram: plotting
temperature vs. luminosities of stars.  Most stars fall along a main
sequence, hotter stars having higher luminosities.

 Luminosity and temperature are related by:
L = sigma T^4 4 pi R^2

So it makes sense that hotter stars have higher luminosities.  There
are hot stars with low luminosities; they must be small; we call these
'white dwarfs'.  Similarly, very luminous red stars must be enormous,
we call them 'red giants' (or for the really big ones, red
supergiants).   For example, Betelgeuse (a red supergiant) has a
radius larger than the orbit of Jupiter, and Sirius B, a white dwarf,
is smaller than Earth.

 We'll see that the Main Sequence is a sequence in mass; hotter stars
are more massive.  (This doesn't hold for non-main-sequence stars,
for which the story is a bit more complicated).   We'll discuss the
measurement of mass of stars next time.

Notes for Lecture 10

© Copyright 2009 Anatoly Spitkovsky and Michael A. Strauss